A general rule of court proceedings is that if one party introduces a person's statements into evidence, the other party generally has the right to cross-examine that witness. There are certain exceptions to this, but the Fifth Amendment does not in and of itself add one; if a defendant doesn't wish to introduce any statements into evidence, the prosecutor may not call him, but the defense may not in general introduce statements by the defendant into the court unless the defendant agrees to take the stand and be subject to cross-examination.
With regard to the case at hand, a fair judge would probably allow some of the statements Zimmerman made to cops to be introduced under a rule called res gestae, which basically suggests that statements made about something that is happening, at the time that it is happening, are more likely to be truthful than statements made later (after a person has had time to make up a lie), but Judge Nelson had shown herself to be sufficiently biased toward the prosecution that I don't think the prosecutor would have had to worry too much about that. The prosecutor's main strategic goal would be to force the defense to decide between leaving out testimony regarding many of George's statements, having George take the stand so as to allow such testimony to be admitted, or getting into a drawn-out argument about which testimony should be admissible without George taking the stand. With the help of Judge Nelson, I think the prosecutor might have been able to put the defense into a situation where they would think their best bet was to put George Zimmerman on the stand. While defendants can and often do benefit from testifying, getting a defendant on a stand can be an opportunity for a skilled prosecutor to trick even (if not especially) the most innocent of defendants into giving evidence against themselves sufficient to score a conviction, even if no other evidence exists.
Personally, I don't like the application of the hearsay rule in this fashion, or even--more broadly--the principle of restricting the kinds of evidence defendants can present, but such rules are pretty well established. I do think it would be entirely proper for the judge to point out to a jury that certain evidence introduced by the defendant does not meet the standards that would generally be required to consider it reliable, and as such the jury would be discouraged from giving it too much weight, but I would posit that a court which forbids the defendant from introducing evidence on the grounds that it is "unreliable" effectively usurps the authority of the jury to decide matters of fact. If the evidence happens to be truthful, hearing it would increase the jury's chances of making a correct decision. If the evidence doesn't happen to be truthful, a jury should be able to figure that out (especially if jurors informed about what should generally be expected for evidence to be considered "reliable").
Do you have a cite to a current Florida statute or case that says the defense cannot call witnesses to testify to a defendant's statements unless the defendant also takes the stand?